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Mirrors > Home > ILE Home > Th. List > triun | Unicode version |
Description: The indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
triun |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliun 3661 | . . . 4 | |
2 | r19.29 2450 | . . . . 5 | |
3 | nfcv 2178 | . . . . . . 7 | |
4 | nfiu1 3687 | . . . . . . 7 | |
5 | 3, 4 | nfss 2938 | . . . . . 6 |
6 | trss 3863 | . . . . . . . 8 | |
7 | 6 | imp 115 | . . . . . . 7 |
8 | ssiun2 3700 | . . . . . . . 8 | |
9 | sstr2 2952 | . . . . . . . 8 | |
10 | 8, 9 | syl5com 26 | . . . . . . 7 |
11 | 7, 10 | syl5 28 | . . . . . 6 |
12 | 5, 11 | rexlimi 2426 | . . . . 5 |
13 | 2, 12 | syl 14 | . . . 4 |
14 | 1, 13 | sylan2b 271 | . . 3 |
15 | 14 | ralrimiva 2392 | . 2 |
16 | dftr3 3858 | . 2 | |
17 | 15, 16 | sylibr 137 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wcel 1393 wral 2306 wrex 2307 wss 2917 ciun 3657 wtr 3854 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-in 2924 df-ss 2931 df-uni 3581 df-iun 3659 df-tr 3855 |
This theorem is referenced by: truni 3868 |
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